Data-driven methods to learn the governing equations of dynamic models have been studied for several decades, see, e.g., [1]}, [2]}, [3]}. Learning linear models from input-output data goes back to Ho and Kalman [4]}. There have been several algorithmic developments for linear systems, for example, the eigensystem realization algorithm (ERA) [5]}, [6]}, and Kalman filter-based approaches [7]}, [8]}, [9]}. Dynamic mode decomposition (DMD) has also emerged as a promising approach to construct models from input-output data and has been widely applied in fluid dynamics applications, see, e.g., [10]}, [11]}, [12]}. Furthermore, there has been a series of developments to learn nonlinear dynamic models. This includes, for example, equations free modeling [13]}, nonlinear regression [14]}, dynamic modeling [15]}, and automated inference of dynamics [16]}, [17]}, [18]}. Utilizing symbolic regression and an evolutionary algorithm [19]}, [20]}, learning compact nonlinear models becomes possible. Moreover, leveraging sparsity (also known as sparse regression), several approaches have been proposed [21]}, [22]}, [23]}, [24]}, [25]}, [26]}. We also mention the work [27]} that learns models using Gaussian process regression. All these methods have particular approaches to handle noise in the data. For example, sparse regression methods, e.g., [21]}, [22]}, [26]} often utilize smoothing methods before identifying models, and the work [27]} handles measurement noise as data represented like a Gaussian process.
